Parametric techniques

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

We also make a distinction between parametric and non-parametric techniques. In the parametric techniques such as linear discriminant analysis, UNEQ and SIMCA, statistical parameters of the distribution of the objects are used in the derivation of the decision function (almost always a multivariate normal distribution... 

The first is to normalize the data, making them suitable for analysis by our most common parametric techniques such as analysis of variance ANOYA. A simple test of whether a selected transformation will yield a distribution of data which satisfies the underlying assumptions for ANOYA is to plot the cumulative distribution of samples on probability paper (that is a commercially available paper which has the probability function scale as one axis). One can then alter the scale of the second axis (that is, the axis other than the one which is on a probability scale) from linear to any other (logarithmic, reciprocal, square root, etc.) and see if a previously curved line indicating a skewed distribution becomes linear to indicate normality. The slope of the transformed line gives us an estimate of the standard deviation. If... 

Parametric techniques based on the multivariate normal (MVN) distribution are particularly well developed. Parameters of the MVN distribution include a covariance or correlation for each pair of variables, as well as a mean and variance for each variable. 

LDA is the first classification technique introduced into multivariate analysis by Fisher (1936). It is a probabilistic parametric technique, that is, it is based on the estimation of multivariate probability density fimc-tions, which are entirely described by a minimum number of parameters means, variances, and covariances, like in the case of the well-knovm univariate normal distribution. LDA is based on the hypotheses that the probability density distributions are multivariate normal and that the dispersion is the same for all the categories. This means that the variance-covariance matrix is the same for all of the categories, while the centroids are different (different location). In the case of two variables, the probability density fimction is bell-shaped and its elliptic section lines correspond to equal probability density values and to the same Mahala-nobis distance from the centroid (see Fig. 2.15A). 

Current methods for supervised pattern recognition are numerous. Typical linear methods are linear discriminant analysis (LDA) based on distance calculation, soft independent modeling of class analogy (SIMCA), which emphasizes similarities within a class, and PLS discriminant analysis (PLS-DA), which performs regression between spectra and class memberships. More advanced methods are based on nonlinear techniques, such as neural networks. Parametric versus nonparametric computations is a further distinction. In parametric techniques such as LDA, statistical parameters of normal sample distribution are used in the decision rules. Such restrictions do not influence nonparametric methods such as SIMCA, which perform more efficiently on NIR data collections. 

Moulines and Laroche, 1995] Moulines, E. and Laroche, J. (1995). Non parametric techniques for pitch-scale and time-seale modification of speech. Speech Communication, 16 175-205. 

The small fraction of data in the tails are probably due to contamination and their elimination removes their disproportionately large influence on hypothesis testing by parametric methods. Non-parametric techniques are relatively insensitive to the data in the tails of the distribution, and unscreened data were used for these analyses. 

Steady-State identification is the first step for data processing in RTO (Bhat and Saraf, 2004) that also includes gross error detection and data reconciliation. In this work we present a review of the techniques used for steady-state identification. But, as literature reports experiences exclusively with parametric methods, we propose the study of some non-parametric techniques. 

In flow analysis, multi-detection is generally accomplished by resorting to multichannel flow analysers, and each channel incorporates a different dedicated detector. Another possibility is to take advantage of multi-parametric techniques such as ICP-OES, anodic stripping voltammetry and UV—visible spectrophotometry with diode array detection. A deeper presentation of this aspect is outside of the scope of this monograph. 

The JCZ3 EOS was the first successful model based on a pair potential that was applied to detonation [11]. This EOS was based on fitting Monte Carlo simulation data to an analytic functional form. Hobbs and Baer [12] have recently reported a JCZ3 parameter set called JCZS. JCZS employs some of the parametrization techniques used in the construction of BKWC. It achieves better accuracy for the detonation of common high explosives than BKW equations of state. Since it is extensively parametrized to detonation, it has difficulty in reproducing reactive shock Hugoniots of hydrocarbons and other liquids [13]. 

In the comparison of calibration methods, the results show that the non-parametric techniques decreased the regression error by approximately 50% over the parametric approaches. The sensor array used in these examples was... 

Quite a variety of different methods can and have been apphed in QSAR work for the evaluation of classification rules [74]. These methods may roughly be divided into two categories, namely parametric or statistical and non-parametric or heuristic techniques. While class separation in the parametric techniques is... 

Parametru/non-parametric techniques This first distinction can be made between techniques that take account of the information on the population distribution. Non parametric techniques such as KNN, ANN, CAIMAN and SVM make no assumption on the population distribution while parametric methods (LDA, SIMCA, UNEQ, PLS-DA) are based on the information of the distribution functions. LDA and UNEQ are based on the assumption that the population distributions are multivariate normally distributed. SIMCA is a parametric method that constructs a PCA model for each class separately and it assumes that the residuals are normally distributed. PLS-DA is also a parametric technique because the prediction of class memberships is performed by means of model that can be formulated as a regression equation of Y matrix (class membership codes) against X matrix (Gonzalez-Arjona et al., 1999). 

In the Svensson model, there are six coefficients Pq, fii, 2. 3. 1 and T2 that must be estimated. The model was adopted by central monetary authorities such as the Swedish Riksbank and the Bank of England (who subsequently adopted a modified version of this model, which we describe shortly, following the publication of the Waggoner paper by the Federal Reserve Bank of England). In their 1999 paper, Anderson and Sleath evaluate the two parametric techniques we have described, in an effort to improve their flexibUity, based on the spline methods presented by Fisher et al. (1995) and Waggoner (1997). 

Anderson and Sleath presented a model in the Bank of England Quarterly Bulletin in November 1999. The main objective of this work was to evaluate the relative efficacy of parametric versus spline-based methods. In fact, different applications call for different methods the main advantage of spline methods is that individual functions in between knot points may move in fairly independent fashion, which makes the resulting curve more flexible than that possible using parametric techniques. In Section 5.5.1 we reproduce their results with permission, which shows that a shock introduced at one end of the curve produces xmsatisfactory results in the parametric curve. 

The outstanding feature of the Anderson-Sleath approach is their adaptation of both spline and parametric techniques. 

Quantitative approaches, based rai a detailed analysis of the product design and the manufacturing processes, demand a lot of informatimi (Nlazi et al. 2006). In parametric techniques, cost is expressed as an analytical function of constituent variables based on statistical methods (NASA). 

What is the significance of these different scales of measurement As was mentioned in Section 1.5, many of the well-known statistical methods are parametric, that is, they rely on assumptions concerning the distribution of the data. The computation of parametric tests involves arithmetic manipulation such as addition, multiplication, and division, and this should only be carried out on data measured on interval or ratio scales. When these procedures are used on data measured on other scales they introduce distortions into the data and thus cast doubt on any conclusions which may be drawn from the tests. Non-parametric or distribution-free methods, on the other hand, concentrate on an order or ranking of data and thus can be used with ordinal data. Some of the non-parametric techniques are also designed to operate with classified (nominal) data. Since interval and ratio scales of measurement have all the properties of ordinal scales it is possible to use non-parametric methods for data measured on these scales. Thus, the distribution-free techniques are the safest to use since they can be applied to most types of data. If, however, the data does conform to the distributional assumptions of the parametric techniques, these methods may well extract more information from the data. 

Data smoothing is a semi-parametric technique, it involves the averaging of the measured dependent parametric variable y. Although smoothing can be seen as a graphical technique, one of its most important advantages is that the dependent variable y does not need to be a specific function of independent variables in time. 

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